The Mathematical Universe

muSome quick words about the mathematical universe, which is the theme of the first chapter of The Big Questions:

1. A “mathematical object” consists of abstract entities (that is, “things” with no intrinsic properties) together with some relations among them. For example, the euclidean plane that you studied in high school geometry consists of points, together with certain relations among them (such as “points A, B and C are collinear”). Mathematical objects can be very complicated. Mathematical objects can have “substructures”, which is a fancy name for “parts”. A line in the plane, for example, is a substructure of the plane.

2. Every modern theory of physics says that our universe is a mathematical object, and that we are substructures of that object. Theories differ only with regard to which mathematical object we happen to be a part of. Particles, forces and energy are not just described by equations; they are the equations (together with abstract, purely mathematical relations among those equations).

3. If you want to think of the universe as something other than a mathematical object — say, something that is controlled by mathematics, or described by mathematics, as opposed to made of mathematics — then you’re up against the fact that nobody has the slightest idea how to construct a useful physical theory along those lines. It’s not just that science rejects all the alternatives; it’s that no scientist (as far as I know) has even been able to imagine a useful alternative. (Perhaps you can find solace in religion.)

4. So — at least if you accept modern physics — at least one mathematical object (namely the one we live in, and are part of) exists physically. This naturally raises the question of which other mathematical objects exist physically.

5. Before you can answer that, you’ve got to ask what it could even mean for one mathematical object to exist physically while another doesn’t. Our universe is a certain mathematical object. Surely there are other mathematical objects that differ from it only in detail. They contain substructures as complicated as we are, in more or less the same ways, and therefore as well equipped as we are to perceive their surroundings as physically real. If you want to claim that our universe is in fact “real” and theirs is not, then you’ve got to explain what that reality consists of.

6. Whatever that reality consists of, it must be some purely mathematical property, because our universe is a purely mathematical object, so that all its properties are purely mathematical.

7. I want to stress that: If, as physics tells us, the universe is “made of math”, then its physical reality is part of that math. We should therefore expect similar mathematical structures to share the property of physical reality.

8. Which structures count as “sufficiently similar” to ours that we should expect them to be physically real? That’s up to us, really, since we’re in the process of defining the term “physical reality”. My own instinct is to call a mathematical object “physically real” if it contains what Max Tegmark calls “self-aware substructures”, i.e. structures that are complicated in a way that makes them aware of their own existence. Tegmark’s own preference is to say that there’s simply no point in discriminating among mathematical objects, and we should simply call all of them physically real.

9. But why bother arguing? Why struggle to come up with a definition for some vaguely imagined notion of “physical existence”? Ockham’s Razor cautions us not to burden ourselves with unnecessary metaphysical baggage. There are mathematical objects. Some have self-aware substructures. Our universe is one of those. The business of science is to figure out which one. What more needs to be said?

10. Finally: I never cease to be amazed by people who uncritically accept the reality of rocks, geese and butterflies but want to deny the reality of mathematical objects. Science tells us that rocks, geese and butterflies are mathematical objects. What else could they be?

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38 Responses to “The Mathematical Universe”


  1. 1 1 improbable

    I think that while the world may indeed be made of mathematics, we are pretty sure that it is NOT ultimately made of the mathematics we actually have in our theories today. In fact we’re not very sure that “ultimately made of” has to have a sensible answer.

    We are sure however that the particles which matter (for the chemistry which matters for life) are very well described by our theories today. We are quite sure that the maths we don’t know doesn’t matter. I think this is enough for your argument: the universe that we experience as living beings is described sufficiently well by maths. So that a simulation of it by this maths would contain us, having this conversation. Nothing else is needed.

    Still the absolutism of #7 (and 4, 5) bothers me. In physics I think fewer people would sign on to this sort of statement now than a few decades ago (thanks largely to Wilson and all that). We have very good low-energy effective field theories, but this is all we have. We have pretty strict upper bounds on their validity, where we are sure that other effects will become important, but which we are are unable to describe. Of course we think that a description of these effects exists, and we think it must be mathematical, but this is professional hope, rather than established knowledge.

  2. 2 2 EricK

    What is the role of time in this whole picture? Equations don’t evolve over time, they just are. However they can describe objects which evolve over time. The universe appears to evolve over time, so the universe can’t be just the equations. Or rather the universe at a point of time can’t be just the equations; however the universe when considered as every possible point of time might be just the equations. So objects which exist within time (like rocks, geese and butterfly) seem to enjoy a completely different kind of existence to the existence enjoyed by equations – the former have a temporary existence, the latter a timeless one. Perhaps this explain your point 10.

  3. 3 3 Philip

    Arguing from ignorance that reality is made of mathematical objects doesn’t convince me. There are very good reasons to doubt the possibility of a direct correspondence between mathematical structures and reality:

    http://www.askphilosophers.org/question/184
    http://www.askphilosophers.org/question/3309

    The most we can claim is that mathematical descriptions of the universe provide approximate descriptions of reality – every physical theory has this character. There is even reason to doubt that a reductionist approach will finally lead to a ToE:

    http://www.pnas.org/content/97/1/28.full.pdf
    http://www.physicsforums.com/showpost.php?p=670934&postcount=7

  4. 4 4 neil wilson

    “Finally: I never cease to be amazed by people who uncritically accept the reality of rocks, geese and butterflies but want to deny the reality of mathematical objects.”

    Finally: I am shocked that people who accept science in so many ways are so convinced that most all of the scientists in the world are completely wrong about Global Warming.

    It amazes me how smart people can pick and choose what they believe to be correct and ignore any evidence that doesn’t support what the person already “knows” to be true.

  5. 5 5 Coupon Clipper

    “Particles, forces and energy are not just described by equations; they are the equations (together with abstract, purely mathematical relations among those equations).”

    SL, can you explain that further? In what sense are particles actually equations?

  6. 6 6 Harold

    You use the terms mathenmatical objects and mathematical structures, are these synonomous?

    IF we can say our universe is a mathematical object, and it has physical existence, then I think we must say that ALL mathematical objects have physical existence.

    The Euclidean plane therefore has a physical existence. I had always thought of it as “describing” a perfect plane, which had no physical existence. Real world physical planes (in our universe) are approximations to this perfect plane.

    If the Euclidean plane has a physical existence, where is it? Is it in a different universe? The “Euclidean plane” universe is too simple to have self aware entities, so it has a physical existence, but nobody will ever be able to perceive it. Is this right? I am struggling a bit here.

  7. 7 7 Thomas Purzycki

    As a fan of quantum immortality, I approve of this theory.

  8. 8 8 Nick

    It could also be that our feeble human minds are simply incapable of describing the universe in a language other than mathematics. For the same reasons we cannot sense extra dimensions, the discovery of a deeper truth might be beyond our brain function. Unsatisfying? Maybe, but so is any theory that relies heavily on conjectures and lacks any sort of predictive power.

  9. 9 9 Steve Landsburg

    Eric K: What is the role of time in this whole picture?

    Think of the simplest example: A world with one dimension of space and one dimension of time. We can represent this world as a plane, with the horizontal axis measuring location and the vertical measuring time. An inhabitant of this world travels along a curve that shows, for each time, the inhabitant’s location.

    To the inhabitant *inside* the model, it feels like the world is “evolving”. But in a view from *outside* the model, there is no evolution. The entire curve exists, all at once.

    So the model is complete, in and of itself, “all at once”.

    Why do the inhabitants in the model perceive time as “evolving”? In this simple model, they probably wouldn’t. IN a more complex model, the inhabitants would have internal structures, like brains, that are shaped by evolution (within the model), and are adapted to perceive things like time evolution because the equations of the model are such that entities with those perceptions are more likely to have descendants, etc. But again, these perceptions are *internal* to the model, and don’t prevent the model from being described, in its entirety, by a single set of equations and relationships.

  10. 10 10 Roger Schlafly

    If you are applying Ockham’s Razor, then it is much simpler to define physical existence in terms of observability, than in terms of self-aware substructures. There are no mathematical objects that have been proved to have any self-aware substructures.

    But then it is simpler still to assume that the world is a figment of my imagination.

  11. 11 11 Jonathan Campbell

    Which (if any) features of the mathematical universe could be derived by a very smart person who had no contact with the outside world, i.e. no empirical evidence to rely on?

  12. 12 12 Jonathan Campbell

    (no empirical evidence other than his experience of his own mind, let’s say)

  13. 13 13 Harold

    “then you’re up against the fact that nobody has the slightest idea how to construct a useful physical theory along those lines.”

    I don’t quite get this. What about Newtons laws of motion? These are pretty useful physical theories, but we know they do not describe the universe, much less are the universe.

  14. 14 14 Steve Landsburg

    Harold: When I said “useful physical theory” I meant “useful theory of the universe” — something like general relativity or quantum field theory, neither of which claims to be complete but each of which is a model of the entire universe.

  15. 15 15 Steve Landsburg

    Harold:

    You use the terms mathenmatical objects and mathematical structures, are these synonomous?

    Yes.

    If the Euclidean plane has a physical existence, where is it? Is it in a different universe? The “Euclidean plane” universe is too simple to have self aware entities, so it has a physical existence, but nobody will ever be able to perceive it. Is this right? I am struggling a bit here.

    Mathematical objects do not have locations. Asking “where is the euclidean plane?” is like asking “where is our universe?”. Our universe contains locations, just as the euclidean plane contains points. The euclidean plane does not live in our universe.

    The answer to the question “does the euclidean plane have physical existence?” depends entirely on what you mean by physical existence. I prefer to say that “physical existence” means that some substructure is self-aware. By that criterion, the plane (almost) surely does not have physical existence. Tegmark prefers to say that *all* mathematical structures have physical existence (which is equivalent to saying that he’s throwing out the notion of “physical existence” entirely, by giving it no independent meaning).

  16. 16 16 EricK

    Is it useful to use the same word – “existence” – for what seems to be two quite different phenomena? Rocks exist but only within a universe. But Universes, and other mathematical structures, just exist (at least according to you and Tegmark).

    You make the point that universes don’t have locations, rather universes contain locations. Couldn’t something similar be said about existence?

  17. 17 17 XiXiDu

    Does a reversal of Greg Egan’s argument against Dust Theory work against the Mathematical Universe?

    “However, I think the universe we live in provides strong empirical evidence against the “pure” Dust Theory, because it is far too orderly and obeys far simpler and more homogeneous physical laws than it would need to, merely in order to contain observers with an enduring sense of their own existence. If every arrangement of the dust that contained such observers was realised, then there would be billions of times more arrangements in which the observers were surrounded by chaotic events, than arrangements in which there were uniform physical laws.” — http://www.gregegan.net/PERMUTATION/FAQ/FAQ.html

    Shouldn’t there be many more logical consistent universes with self-aware substructures that do not feature a physical-chemical-biological evolution? Why do we seem to be in a universe with qualities this chaotic and unnecessary to merely contain self-aware substructures?

  18. 18 18 Rhys

    Steve, Take step 10 of your article
    Replace the words “mathematical object” with “god” and replace the word “science” with “the bible”. Are you aware of the irony in your article?

  19. 19 19 Super-Fly

    I realize I’m opening a can of worms here (and I’m only about half-serious) but can we say that God does exist and the Christian Bible is infallible… just not necessarily in our universe? I suppose it wouldn’t be very helpful then, but it would still be interesting.

    On a slightly less tongue-in-cheek note, in TBQ, you kinda lost me with the concept of consciousness. Surely this mathematical pattern of ours (namely, our world, DNA, etc.) is contained in other patterns. So… what about consciousness? Is there another conscious entity with all of my properties? What makes *this* me special? Why am I the only one who feels things?

  20. 20 20 AC

    Prof. Landsburg writes as if this view is self-evident from the physics. Would most physicists agree then? I suspect not.

  21. 21 21 AC

    To follow-up on above, I am not arguing by authority, I am just curious about how physicists would approach this argument.

  22. 22 22 Neil

    Forgive the resort to authority, but I think I will go with Einstein on this one:

    “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

  23. 23 23 Steve Landsburg

    AC:

    Would most physicists agree then?

    I am not a physicist, and I don’t hang around with physicists the way I (sometimes) do with mathematicians, so I am not sure I am right here. But what I think most physicists would agree to is:

    a) All modern cosmological theories, taken literally, say that the universe is a mathematical object and
    b) We probably shouldn’t take those theories quite so literally.

    Part of my response is that if you let yourself pick and choose which parts of your theories to take literally, then you’re allowing yourself a dangerous amount of leeway to believe whatever you want. My guess is that most physicists would understand what I’m saying but would not necessarily buy into it.

  24. 24 24 Cullen Roberts

    If all mathematical objects exist physically, then the ontological proof for the existence of God must be taken seriously.

  25. 25 25 Steve Landsburg

    Cullen Roberts:

    If all mathematical objects exist physically, then the ontological proof for the existence of God must be taken seriously.

    Only if God is a mathematical object.

  26. 26 26 Seth

    So a mathematical object goes from consisting of abstract entities to real entities when there’s a self-aware substructure?

  27. 27 27 Steve Landsburg

    So a mathematical object goes from consisting of abstract entities to real entities when there’s a self-aware substructure?

    Seth: That’s my preferred way to think of it. Tegmar, I think, would prefer to say that there’s no point in distinguishing between the abstract and the real.

  28. 28 28 Robert

    “If you want to think of the universe as something other than a mathematical object — say, something that is controlled by mathematics, or described by mathematics, as opposed to made of mathematics — then you’re up against the fact that nobody has the slightest idea how to construct a useful physical theory along those lines.”

    What is the distinction here?

  29. 29 29 Alan Wexelblat

    I see that others have jumped in. Since I’m not myself well versed in this area, I asked someone who is – a friend of mine with a PhD in physics from Caltech who has worked for NASA as an experimental physicist for quite some time. I quote his response, with permission:

    “[M]ost physicists believe that the universe can mostly be described by math (self-awareness _might_ be an exception), and would buy the fact that there might be a mathematical object that is “equivalent” to the universe, but does that mean that the universe is “only” a mathematical object? No.

    “And things like the Heisenberg uncertainty principle make it such that the simplest way one can fully describe the universe needs something at least the size of the universe. And it’s quite a jump to claim that because one might conceive (but can’t prove) that similar mathematical objects might exist, they are just as real as the one that is equivalent to this universe. There might be a few theoretical physicists who think this way, but nearly all experimentalists, and I think most theoreticians, are more grounded in the fact that the universe is observable, and thinking about unobservable possibilities such as similar mathematical objects might be fun, but is philosophy and not physics.”

  30. 30 30 Harold

    “you’re up against the fact that nobody has the slightest idea how to construct a useful physical theory along those lines”…
    …yet. Perhaps we have a long way to go.

    If our universe were to have an existence not AS a mathematical object, but described by mathematics, would we be able to tell?

    Newtonian mechanics is a description of a simple universe. I suppose this is much the same as the Euclidean plane. A universe described by Newtonian mathematics would have an existence, with all conceivable combinations of solid objects interacting through gravity. This universe could never contain self-aware entities because it does not contain explanations of other interactions, so cannot contain them. You would say that this has no physical existence, but Tegmar would say it does (sort of).

    I am seeking the distinction here. For a while, Newton was the best description we had for the universe, recognisably incomplete, but the first real explanation of the movements of the planets (and everything else). It could be described as a model of the entire universe. From the perspective of a person who knows only Newtonian mechanics:

    1) would they have said that the universe WAS a mathematical object?
    2) If not, do you think they should have done?
    3) If not, then what is it about our current theories that makes them different?

    On a different note, do you think that the universe we inhabit is like the one with a single dimension of space and time you described earlier? This was described over all time with a single set of equations, and was completely deterministic.
    Chaos theory has shown us that the state after time is dependent precisely on the starting condititons. There is more possible variety in the starting conditions than can be “coded” for within the universe. Is this the origin of an infinity of universes? Each one is completely determined from start to finish by the original conditions?

  31. 31 31 XiXiDu

    What if Current Foundations of Mathematics are Inconsistent? http://video.ias.edu/voevodsky-80th

  32. 32 32 Thomas Purzycki

    I’ve had some time to mull this over; here are my thoughts as I tried to follow the idea to logical conclusions:

    So if I am made of maths, then as a mathematical substructure, there must be a number of structures (universes) capable of including me, right? Furthermore, as a mathematical substructure, it does not seem to be meaningful to say that substructures identical to me in other universes are copies of me. Since a substructure can be plugged into any appropriate structure and exists independently of those structures, it doesn’t seem like a stretch to say that all of those identical substructures are me and that I am that substructure in all of its locations simultaneously.

    Unless I’m dreaming or under the influence of drugs, my subjective reality consists of a well ordered universe where the laws of physics are surprisingly consistent in the sense that they seem to be constant regardless of location and time. I perceive time in a linear fashion and it seems obvious that there are certain rules that govern how the mathematical substructure that is my consciousness evolves (if a change breaks those rules – I could either no longer be considered conscious, or no longer be considered me). Clearly, I’m not going to experience a universe that evolves such that those rules are broken. However, it is not so clear to me why the universe I experience is so well behaved.

    When I go to bed at night, my consciousness’s substructure evolves to the point where it becomes very probable that my next experience will be waking up in the morning (assuming dreamless sleep). In a sense, my consciousness’s substructure disappears from one set of universes and reappears as me waking up in another set of universes. For some reason, every time I wake up, the sky is still blue, Polaris still points north, and I still have to pay taxes. Apparently, the the number of universes I can wake up in that contain those constraints far outnumber the universes I can wake up in that do not have those constraints, otherwise I’d expect to experience changes like that more often. Why is this? I’m simplifying a bit with the waking up part – presumably in a mathematical universe, your subjective experience would have a chance of being diverted into one of these alternate realities at any given instant.

  33. 33 33 Steve Landsburg

    Thomas Purzycki:

    1) A point on a circle and a point on a line have exactly the same structure (namely no structure at all); nevertheless we don’t normally think of them as the same point.

    2) The issues that are troubling you do not seem unique to the notion of a mathematical universe. The same issues arise in many interpretations of quantum mechanics, including all of those that seem to work best for cosmology (many-worlds or many-histories). Even if every quantum event causes the universe, including yourself, to follow multiple branches “simultaneously”, you have the psychological sensation of being present in only one of them. That observation might not resolve your issue, but it does indicate that you can’t avoid it by rejecting the notion of a mathematical universe.

    3) Max Tegmark has proposed that in principle, the way we should make predictions in physics is by averaging outcomes in all of the various universe that can contain us (or identical copies of us). So he is certainly prepared to take seriously the idea that we in some sense inhabit many universes at once.

  34. 34 34 Scott H.

    I dunno…

    If this is all about redefining your terms, maybe you aren’t really saying that much here.

    The real questions would be: are the mathematical models good? are they beautiful? are they sublime?

    Doesn’t the object answer have to be: yes!

  35. 35 35 ted

    My undesrstanding is that physics maps some phenomena to mathematical objects – the big universe to a 4-D psuedo Riemannian manifold, free elementary particles to irreducible representations of the Poincare group, but that there is no comprehensive mapping that works for everything.

    So what mathematical object is the universe?

  36. 36 36 Steve Landsburg

    Ted:

    So what mathematical object is the universe?

    Answering this question amounts to constructing the right “theory of everything”. In other words, it’s a major research question.

  37. 37 37 Seth

    Steve: If a self-aware substructure makes a mathematical object real, what’s a dream where you become self-aware?

  38. 38 38 n+1th iteration

    I apologize if I’m beating a dead horse here.
    I see your argument taking two phases: first, you establish that modern physical theories say everything is a mathematical object, and then you make an appeal to elegance, or at least ockham’s razor, for why mathematical objects are all we need in our cosmology.

    That makes a lot of sense. If the gremlins running the universe make everything follow certain mathematical patterns, then the gremlins are worthless: we are best served by believing that mathematical objects are the source of the pattern, and that the gremlins are a lie.

    But it’s this idea that everything is a ‘mathematical object’ that keeps sticking in my throat. Pardon my anecdote, but I’ve had one of my math professors describe math as the study of ‘necessary implications’: no matter what situation you’re in, if you have any sort of object, and you accept that it satisfies the conditions for it to be, say, a field, then you must accept everything mathematics tells you about that object–the behavior of its elements, its algebraic extensions, everything. Mathematicians simply take sets of rules, and look for their implications. We say things are isomorphic, or homeomorphic, i.e. that they are effectively the same all the time. It’s only in cases where we’re picking from a constructed set of objects (the integers, etc.) that we end up claiming two things are actually the same thing.

    I’m not a physicist, so I’ll take you on your word that modern physical theories insist that space really is a mathematical object, as opposed to just behaving like one. But the ‘mathematical object’ at the bottom of the analysis is usually defined like “something that satisfies these properties.”

    Please clarify if I’m no longer following what you said, but that result seems kind of trivial. Another math professor of mine tried to sum up isomorphisms by saying “If it walks like a duck, and quacks like a duck, then it is a duck.” But if we conclude that reality is composed of mathematical objects–perhaps that our reality is made up of ducks–do we end up saying anything other than “reality is made up of things that walk and quack like ducks”?

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